Shady Schach

The Gemorah in Sukkah (8b) says תָּנוּ רַבָּנַן: גַּנְבַּ״ךְ: סוּכַּת גּוֹיִם, סוּכַּת נָשִׁים, סוּכַּת בְּהֵמָה, סוּכַּת כּוּתִים סוּכָּה מִכׇּל מָקוֹם  כְּשֵׁרָה, וּבִלְבַד שֶׁתְּהֵא מְסוּכֶּכֶת כְּהִלְכָתָהּ מַאי ״כְּהִלְכָתָהּ״? אָמַר רַב חִסְדָּא: וְהוּא שֶׁעֲשָׂאָהּ לְצֵל סוּכָּה.  Rashi explains אמר רב חסדא - האי כהלכתה דקאמר הוא שמסוככת יפה דמוכחא מלתא שעשייתה הראשונה לצל היתה ולא לצניעות בעלמא דאע"ג דסוכה לשם חג לא בעינן לשם סוכה בעינן ולצל הוא דמיקריא סוכה שסוככת מן החורב.  The simple understanding of this law would be that the definition of a sukkah is to give shade (the schach is the sukkah as Rashi says at the beginnig of the masechta,) and therefore the schach must be placed with intention of giving shade.  However, this is difficult for according to Rebbe Yehuda a sukkah is valid even if the schach is placed very high up where it is not producing any shade as the Gemorah in the beginning of the masechta says?  Furthermore, even according to the Rabbanan one is allowed to place a sukkah in between mountains where the schach is not producing shade, how is such a sukkah valid?  And what is the peshat in the Gemorah itself on 8b, if there is such a condition that a sukkah must be made with intention of shade then what is the  chiddush of Rav Chisdah in saying the sukkah must be made כהלכתה, obviously all the rules of making a sukkah must be followed?   

What we see is that there are two ways to make a sukkah.  One way is to intentionally make a sukkah for the holiday.  In that manner one does not need to make the sukkah for the sake of shade.  It is only when the sukkah is not made for the sake of the holiday that the sukkah must be made to provide shade.  That is the chiddush of Rav Chisdah.  In such a situation when the sukkah was not made for the sake of the holiday then there is a condition that the sukkah must have been made for the sake of shade.  We see this from Rashi 2b that says בד"ה לימות המשיח - אבל סוכת מצוה אינה לצל.  Rashi says the sukkah of the mitzvah does not need to provide shade.  According to this approach shade is not defining what a sukkah is, is it a condition in the making of a sukkah that in order to demonstrate that it is a sukkah of mitzvah it must provide shade.  However, it needs to be explained why we would assume this is a law in the making of a sukkah, וצ"ב.  (Based upon a shiur by Rabbi Moshe Brown.)